So I'm working through Matilde Marcolli's monograph on Arithmetic Noncommutative Geometry, particularly Chapter 3 "Quantum Statistical Mechanics and Galois Theory" (That's also the title of my 4th year project with Professor Kim at UCL.) Due to ongoing personal and financial problems, I don't have nearly as much time to be doing maths as I should like, but I was able to do some reading today, and I thought I should write something done to denote what I think I might understand [not much!] and what I definitely do not understand [most of it!]

I was able to take a first glance at the short introduction chapter as well as the meat-and-potatoes in Chapter 3. Here's what I know about non-commutative geometry so far:

Non-commutative geometry tries to extend the tools and methods of ordinary geometry [I'm not sure what "ordinary geometry" is, to be honest] to work on nasty quotient spaces where the "Ring of functions" [I believe that's related to Morse theory?] is too small in some sense. Particularly, if X is a space and ~ an equivalence relation on it, then the "Ring of functions" for the quotient X/~ is defined as a subset of the ring of functions of X where each function is invariant on the induced partitions. From what I know about quotient spaces, it seems very unlikely that the quotient ring of functions is anything but a collection of boring constance functions. But apparently, and I don't understand how at all, we can use a non-commutative algebra of coordinates (much like using non-commuting variables in relativistic quantum mechanics) to get an interesting ring of functions, and this is somehow related to an extension of the Gel'fand-Naimark theorem (Marcolli mentions this as "X is a locally compact Hausdorff space <=> C_0(X) is an abelian C*-algebra, but looking the names up in Arveson's "An invitation to C*-algebras" says that the theorem says "Every abstract C* algebra with identity is isometrically *-isomorphic to a C*-algebra of operators", I think I have an idea what that theorem means - its a bit like saying any linear operator on a finite vector space can be expressed as a matrix...I think). I don't know how the two are related, if at all.

I'm interested in non-commutative geometry because it relates spontaneous symmetry breaking in a quantum mechanical system to $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$. So lets talk about what I know of this relationship:

Connes, et al, constructed quantum statistical mechanical systems that recover the class field theory of $\mathbb{Q}$ and imaginary fields, and possibly even quadratic fields (I don't know what class field theory is, but its apparently related to $Gal(\mathbb{Q}^{cycl} / \mathbb{Q})$ as best I can tell.) A quantum statistical mechanical system is a C* dynamical system, which is a C* algebra A together with a family of automorphism $\sigma_t$ of A that can be described with a single parameter: t for time. The construction of system has something to do with $\mathbb{Q}$-lattices, which is a lattice $\Lambda$ of $\mathbb{R}^n$ together with a homomorphism $\phi : \mathbb{Q}^n / \mathbb{Z}^n \to \mathbb{Q} \Lambda / \Lambda$ (I have no idea what that means.) Two $\mathbb{Q}$-lattices are commensurable if their torsion labels interact in some way I don't understand. Commesurability is an equivalence relation, and the C* algebra on the set of commensurability classes of $\mathbb{Q}$-lattices in $\mathbb{R}$ is the Bost-Connes system.

I think my goal for next couple of weeks is to understand that construction.

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